Decomposition numbers of the cyclotomic Brauer algebra over the complex field, II

TL;DR

This paper computes the decomposition numbers of the cyclotomic Brauer algebra over the complex numbers, expressing them through parabolic Kazhdan-Lusztig polynomials of type D under certain conditions.

Contribution

It provides a new method to determine decomposition numbers of the cyclotomic Brauer algebra using Kazhdan-Lusztig polynomials, linking algebraic and combinatorial structures.

Findings

Decomposition numbers expressed via Kazhdan-Lusztig polynomials

Established connection under Condition 1.2

Applicable to arbitrary parameters over complex field

Abstract

Following Nazarov's suggestion, the cyclotomic Nazarov-Wenzl algebra is referred to as the cyclotomic Brauer algebra. This paper focuses on computing the decomposition numbers of the cyclotomic Brauer algebra over $\mathbb{C}$ with arbitrary parameters. We show that these decomposition numbers can be expressed in terms of the parabolic Kazhdan-Lusztig polynomials of type $D_n$, with a parabolic subgroup of type $A$, under Condition 1.2.